Rational difference equations with positive equilibrium point
Bull. Korean Math. Soc. 2010 Vol. 47, No. 3, 645-651
Published online May 1, 2010
Art\= uras Dubickas
Vilnius University
Abstract : In this note we study positive solutions of the $m$th order rational difference equation $x_{n}=(a_0+\sum_{i=1}^m a_ix_{n-i})/(b_0+\sum_{i=1}^m b_ix_{n-i}),$ where $n=m,m+1,m+2,\ldots$ and $x_0,\ldots,x_{m-1}>0.$ We describe a sufficient condition on nonnegative real numbers $a_0, a_1,\ldots,a_m,b_0,b_1,\ldots,b_m$ under which every solution $x_n$ of the above equation tends to the limit $(A-b_0+\sqrt{(A-b_0)^2+4a_0B})/2B$ as $n \to \infty,$ where $A=\sum_{i=1}^m a_i$ and $B=\sum_{i=1}^m b_i.$
Keywords : difference equations, equilibrium point, convergence of sequences, upper and lower limits
MSC numbers : 39A11, 40A05
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